2 emf problems

1] A square loop of wire, with sides of length 'a' lies in the first quadrant of the xy-plane, with one corner at the origin. In this region there is a non-uniform time-dependent magnetic field [itex]\vec B (y,t) = ky^3t^2\hat z[/itex]. Find the induced emf in the loop.

2] A perfectly conducting spherical shell of radius 'a' rotates about the z-axis with angular velocity [itex]\omega[/itex] in a uniform magnetic field [itex]\vec B = B_0\hat z[/itex]. Calculate the emf developed between the "north pole" and the equator.

1) What is [itex]\vec A[/itex]? From your equation, I think you mean it to be the vector area of the surface. You have to be careful, since the field is not uniform, it depends strongly on y. So the flux through the square is not simply the product of B and the area A. You have to integrate over the surface to find the flux at a certain instant of time:

ONLY when [itex]\vec B[/itex] does not depend on position can you remove it from the integral and then:
[tex]\Phi=\int \vec B \cdot d\vec a=\vec B \cdot \int d\vec a=\vec B \cdot \vec A[/tex]

2) This problem is a bit different. You have a conducting surface rotating a magnetic field, so there is a magnetic force acting on each surface element, given by the Lorentz force law. The integral of this force per unit charge gives the emf.

1) What is [itex]\vec A[/itex]? From your equation, I think you mean it to be the vector area of the surface. You have to be careful, since the field is not uniform, it depends strongly on y. So the flux through the square is not simply the product of B and the area A. You have to integrate over the surface to find the flux at a certain instant of time:

Yes, I should have been more careful (since [itex]\vec A[/itex] could also mean the magnetic potential.)

[tex]\Phi(t)=\int \limits_{\square}\vec B \cdot d\vec a[/itex].
Then find [itex]\varepsilon[/itex] by taking the time-derivative.
ONLY when [itex]\vec B[/itex] does not depend on position can you remove it from the integral and then:
[tex]\Phi=\int \vec B \cdot d\vec a=\vec B \cdot \int d\vec a=\vec B \cdot \vec A[/tex]

2) This problem is a bit different. You have a conducting surface rotating a magnetic field, so there is a magnetic force acting on each surface element, given by the Lorentz force law. The integral of this force per unit charge gives the emf.

1) Looks good.
2) [itex]\vec f \times \vec v=\omega rB_0 \hat r[/itex] is correct, but note that the direction is pointing away from the axis of rotation and that r is the distance from this axis (cylindrical coordinates as opposed to spherical).